A Fast Spectral Algorithm for Mean Estimation with Sub-Gaussian Rates

by   Zhixian Lei, et al.

We study the algorithmic problem of estimating the mean of heavy-tailed random vector in R^d, given n i.i.d. samples. The goal is to design an efficient estimator that attains the optimal sub-gaussian error bound, only assuming that the random vector has bounded mean and covariance. Polynomial-time solutions to this problem are known but have high runtime due to their use of semi-definite programming (SDP). Conceptually, it remains open whether convex relaxation is truly necessary for this problem. In this work, we show that it is possible to go beyond SDP and achieve better computational efficiency. In particular, we provide a spectral algorithm that achieves the optimal statistical performance and runs in time O(n^2 d ), improving upon the previous fastest runtime O(n^3.5+ n^2d) by Cherapanamjeri el al. (COLT '19) and matching the concurrent work by Depersin and Lecué. Our algorithm is spectral in that it only requires (approximate) eigenvector computations, which can be implemented very efficiently by, for example, power iteration or the Lanczos method. At the core of our algorithm is a novel connection between the furthest hyperplane problem introduced by Karnin et al. (COLT '12) and a structural lemma on heavy-tailed distributions by Lugosi and Mendelson (Ann. Stat. '19). This allows us to iteratively reduce the estimation error at a geometric rate using only the information derived from the top singular vector of the data matrix, leading to a significantly faster running time.


page 1

page 2

page 3

page 4


Fast Mean Estimation with Sub-Gaussian Rates

We propose an estimator for the mean of a random vector in R^d that can ...

Mean Estimation with Sub-Gaussian Rates in Polynomial Time

We study polynomial time algorithms for estimating the mean of a heavy-t...

Algorithms for Heavy-Tailed Statistics: Regression, Covariance Estimation, and Beyond

We study efficient algorithms for linear regression and covariance estim...

A spectral algorithm for robust regression with subgaussian rates

We study a new linear up to quadratic time algorithm for linear regressi...

Quantum Sub-Gaussian Mean Estimator

We present a new quantum algorithm for estimating the mean of a real-val...

Robust Gaussian Covariance Estimation in Nearly-Matrix Multiplication Time

Robust covariance estimation is the following, well-studied problem in h...

Multi-Normex Distributions for the Sum of Random Vectors. Rates of Convergence

We build a sharp approximation of the whole distribution of the sum of i...